The following are the results of a principal component analysis, on Z, of data collected from a fruit fly experiment attempting to relate a measure of fly activity, WFB = wing beat frequency, to the...

The following are the results of a principal component analysis, on Z, of data collected from a fruit fly experiment attempting to relate a measure of fly activity, WFB = wing beat frequency, to the chemical activity of four enzymes, SDH, F UM, GH, and GO. Measurements were made on n = 21 strains of fruit fly. (Data courtesy of Dr. Laurie Alberg, North Carolina State University.)

(a) Compute the proportion of the dispersion in the X-space accounted for by each principal component. (b) Compute the condition number for Z and the condition index for each principal component. What do the results suggest about possible variance inflation from collinearity?

(c) Describe the first principal component in terms of the original centered and standardized variables. Describe the second principal component.

(d) The sum of the variances of the estimates of the least squares regression coefficients, tr[Var(β)] = Σ(1/λ_j)σ², must be larger than σ₂/λ₄. Compute this minimum (in terms of σ²). How does this compare to the minimum if the four variables had been orthogonal?